Empty sum

In mathematics, an empty sum, or nullary sum is a summation where the number of terms is zero. The natural extension of a non-empty sum is to set the value of any empty sum of numbers to the additive identity. In mathematics, an empty sum, or nullary sum is a summation where the number of terms is zero. The natural extension of a non-empty sum is to set the value of any empty sum of numbers to the additive identity. If a1, a2, a3,... is a sequence of numbers, and is the sum of the first m terms of the sequence, then for all m = 1,2,... provided that the following conventions are used: s 1 = a 1 {displaystyle s_{1}=a_{1}} and s 0 = 0 {displaystyle s_{0}=0} . In other words, a 'sum' s 1 {displaystyle s_{1}} with only one term evaluates to that one term, while a 'sum' s 0 {displaystyle s_{0}} with no terms evaluates to 0. Allowing a 'sum' with only 1 or 0 terms reduces the number of cases to be considered in many mathematical formulas. Such 'sums' are natural starting points in induction proofs, as well as in algorithms. For these reasons, the 'empty sum is zero extension' is standard practice in mathematics and computer programming. For the same reason, the empty product is taken to be the multiplicative identity. For summations defined in terms of addition of other values than numbers (such as vectors, matrices, polynomials), in general of values in some given monoid, the value of an empty summation is taken to be its identity element. The notion of an empty sum is useful for the same reason that the number zero and the empty set are useful: while they seem to represent quite uninteresting notions, their existence allows for a much shorter mathematical presentation of many subjects. In linear algebra, a basis of a vector space V is a linearly independent subset B such that every element of V is a linear combination of B.Similar to the empty sum extension, the zero-dimensional vector space V={0} has a basis, namely the empty set. Consider monoids of numeric sequences S {displaystyle S} (with concatenation operation and 0-length sequence ε as the identity element, also known as Kleene star)and of numeric multisets M {displaystyle M} (with multiset union operation and empty set as the identity element).Since the additive numeric monoids (whether integers, reals etc) are associative, the sum of a numeric sequence is well defined.Since the numeric monoids are also commutative, the sum of a numeric multiset is also well defined.